Domanda del libro di testo
Identify each equation without completing the square.
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 8, Problema 49
Graph each ellipse and give the location of its foci. 9(x − 1)²+4(y+3)² = 36
Guida verificata passo dopo passo1
Rewrite the given equation in standard form by dividing both sides by 36: \(\frac{9(x - 1)^2}{36} + \frac{4(y + 3)^2}{36} = \frac{36}{36}\).
Simplify the fractions to get: \(\frac{(x - 1)^2}{4} + \frac{(y + 3)^2}{9} = 1\).
Identify the values of \(a^2\) and \(b^2\) from the denominators. Here, \(a^2 = 9\) and \(b^2 = 4\). Since \(a^2 > b^2\), the major axis is vertical.
Find the center of the ellipse, which is at the point \((h, k) = (1, -3)\) from the terms \((x - h)^2\) and \((y - k)^2\).
Calculate the focal distance \(c\) using the formula \(c^2 = a^2 - b^2\). Then, determine the coordinates of the foci by moving \(c\) units along the major axis (vertical) from the center.
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Standard Form of an Ellipse
An ellipse equation in standard form is written as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center. The values a² and b² represent the squares of the semi-major and semi-minor axes, respectively. Converting the given equation to this form is essential for graphing and analyzing the ellipse.
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5:12
Graph Ellipses at Origin
Identifying the Major and Minor Axes
The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. By comparing a² and b², you determine which axis is major (larger value) and which is minor (smaller value). This helps in correctly sketching the ellipse and locating its vertices.
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05:01Identifying Intervals of Unknown Behavior
Foci of an Ellipse
The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their locations are found using c² = |a² - b²|, where c is the distance from the center to each focus along the major axis. Knowing the foci is crucial for understanding ellipse properties.
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5:30Foci and Vertices of an Ellipse
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